Analysis and finite element approximation for nonlinear problems in poroelasticity and ...

119 Pages·2013·1.19 MB·English

Analysis and finite element approximation for nonlinear problems in
poroelasticity and bioconvection
by
Song Chen
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
May 5, 2013
Keywords: Nemytskii operator, poroelasticity, bioconvection, Rothe’s method
Copyright 2013 by Song Chen
Approved by
Yanzhao Cao, Chair, Professor of Mathematics and Statistics
A. J. Meir, Co-chair, Professor of Mathematics and Statistics
Wenxian Shen, Professor of Mathematics and Statistics
Georg Hetzer, Professor of Mathematics and Statistics
Abstract
This dissertation is concerned with nonlinear systems of partial differential equation
with solution dependent physical coefficients satisfying the Nemytskii assumption. Such
equations arise from two important application fields: poroelasticity and bioconvection.
First, we consider a quasi-static poroelasticity model with dilatation dependent hy-
draulic conductivity and an implicit time derivative. We derive the existence and unique-
ness of solutions using the modified Rothe’s method, Brouwer’s fixed point Theorem and the
Sobolev embedding Theorem. Next we construct a finite element approximation with linear
elements and establish the optimal error estimate. We then conduct numerical examples to
verify the convergence and simulate the diffusion in a fluid saturated sponge.
Second, we study the bioconvection model, a coupled Navier-Stokes type equation, with
concentration dependent viscosity. We combine the theory of the Navier-Stokes equation
and the modified Rothe’s method to establish existence and uniqueness of solutions of both
steady and time dependent bioconvection. After that we perform finite element analysis with
Taylor-Hood elements and prove the convergence theorem. Finally numerical examples are
constructed using lab data to verify the convergence of the numerical scheme and simulate
the convection pattern formed by micro-organisms inside a culture fluid.
ii
Acknowledgments
It would have been impossible to finish my dissertation without the guidance of my
committee members and support from my family and friends. First of all, I would like to
gratefully and sincerely thank my adviser, Dr. Yanzhao Cao, for his research guidance,
passion, caring and patience. Not only did Dr. Cao provide all the advices and directions
as a professional mathematician, he also shared with me his valuable life experiences to
inspire my life and career, as a wise elder and a close friend. I also want to thank my co-
adviser, Dr. A. J. Meir, for his speciality in scientific computing, my committee members Dr.
Wenxian Shen and Dr. George Hetzer for guiding my research for the past several years and
especially Dr. Xing Fang who was kind enough to be the university reader of my dissertation
and provided many valuable advices on my defence. Finally, I want to thank my parents
and my lovely wife, Yilanna Hu, who have been there supporting me, now and always.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Poroelasticity with dilatation dependent hydraulic conductivity . . . . . . . 2
1.2 Bioconvection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Plan of dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Notations and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Steady poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Existence and uniqueness of a weak solution of quasi-static poroelasticity . . 16
2.3 Numerical approximation with the finite element method . . . . . . . . . . . 33
2.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3 Steady bioconvection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Existence and uniqueness of a weak solution . . . . . . . . . . . . . . . . . . 51
3.2 Numerical approximation using the finite element method . . . . . . . . . . . 60
3.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4 Time dependent bioconvection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Existence and uniqueness of a solution . . . . . . . . . . . . . . . . . . . . . 73
4.2 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
iv
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
v
List of Figures
2.1 The plot of errors for a fixed time step . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 The plot of errors for time step k = h2 . . . . . . . . . . . . . . . . . . . . . . . 47
2.3 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.4 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1 Concentration and velocity field for α = 1% . . . . . . . . . . . . . . . . . . . . 70
3.2 Concentration and velocity field for α = 20% . . . . . . . . . . . . . . . . . . . 71
3.3 Concentration and velocity field for α = 20% with constant viscosity . . . . . . 71
3.4 Concentration and velocity field for α = 30% . . . . . . . . . . . . . . . . . . . 72
vi
List of Tables
2.1 The converge rate for a fixed time step in the L2(I;L2(Ω)) norm . . . . . . . . . 45
2.2 The converge rate for a fixed time step in the L2(I;H1(Ω)) norm . . . . . . . . 46
0
2.3 The converge rate for time step k = h2 in the L2(I;L2(Ω)) norm . . . . . . . . . 46
2.4 The converge rate for time step k = h2 in the L2(I;H1(Ω)) norm . . . . . . . . 46
3.1 Convergence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.2 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1 Convergence rate in L2(I;L2(Ω)) . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2 Convergence rate in L2(I;H1(Ω)) . . . . . . . . . . . . . . . . . . . . . . . . . . 104
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Chapter 1
Introduction
This dissertation is devoted to the study of the well posedness and numerical approxi-
mations of systems of partial differential equation (PDE) with solution dependent physical
parameters arising from two application fields: poroelasticity and bioconvection.
In mathematical modeling, we usually consider a linear system under certain ideal ho-
mogeneous assumptions about the physical parameters. In practice, however, the physical
parameters are generally inhomogeneous and related to the solution of the system of PDE.
A generic PDE with solution dependent physical coefficient can be written as
(cid:16) (cid:17)
L x,t,u(t,x),λ(t,x,u) = f , x ∈ Ω, t > 0, (1.1)
where Ω is the physical domain, t refers to time, L denotes a general differential operator
and λ(t,x,u) is the physical parameter depending on the solution u. When L is a linear
differential operator, (1.1) is often reduced to a quasi-linear PDE. Theoretical analysis of
this problem concerning the well posedness and regularity of solutions can be found in [1],
[2], and [3]. Finite element approximations of some simple quasi-linear elliptic PDEs and
parabolic PDEs (1.1) are presented in [4], [5] and [6].
In this dissertation, we consider two mathematical models taking the form of coupled
PDEs with solution dependent coefficients. The first model arises in poroelasitcity with di-
latationdependenthydraulicconductivityandthesecondmodelisaboutbioconvectionswith
concentration dependent viscosity. Though the physical backgrounds of the two problems
are quite different, we will use similar approaches to study the well posedness and numerical
approximations of the two underlying PDE systems. For instance we will use the modified
1
Rothe’s method to study the existence of weak solutions of both of the two evolution PDE
systems. We will use the fact that the physical parameters in both of the two models give
rise to Nemiskii operators to derive the well-posedness results under minimum regularity
assumptions on the input data.
1.1 Poroelasticity with dilatation dependent hydraulic conductivity
Motivation. We are surrounded by porous elastic solid materials: natural (e.g., rocks,
soils, shale, living tissue, the brain, the heart) and man-made (e.g., cement, concrete, filters,
foams, ceramics, gels, clays). Porous material has a solid matrix structure with small pores
inside which contain air or fluid. Because of their ubiquity and unique properties, porous
materialsareofgreatinteresttonaturalscientistsandengineers([7,8,9,10]). Porousmedia
finds applications in diverse areas include reservoir engineering [11], biomechanics [12, 13, 14]
and environmental engineering [15, 16, 11].
Mathematical model. Poromechanics is a branch of physics and specifically contin-
uum mechanics and acoustics that studies the behaviour of fluid-saturated porous media.
The particular mathematical model we are interested in describes the swelling and shrinking
of an elastic deforming porous medium coupled with the fluid.
Due to the elastic nature of the porous medium, the study of fluid saturated porous
media is called poroelasticity. Terzaghi [17] first derived a one dimensional model in soil
mechanics to involve the influence of the fluid inside a solid body. The model was later
extended to three dimension by Rendulik [18]. A mathematical formulation is derived in
Biot’s work [19] and studies in his other work between 1955 and 1962 [20, 21, 22, 23, 24],
which are later considered to be the foundation of modern poroelasticity theory.
2
The following equations are the essence of Biot’s poroelasticity theory [19] (details of
the theory can be found in [25]) in which the coupled constitutive equations takes the form
 ∂
 (momentum)+stress = external force,

∂t
(1.2)
∂

 (fluid content)+flux = external fluid source.
∂t
Next we convert (1.2) into a coupled system of partial differential equations for the fluid
pressureandthedisplacementoftheporoelasticmedium. Tothisend,wedenotethepressure
by p and the displacement by u. According to Hooke’s law in 3-D the effective stress caused
by the deformation of the solid matrix is given by
τ = 2µε+λtr(ε)I. (1.3)
e
1
Here ε = (∇u + ∇uT) is the strain, tr(ε) is the trace of ε, I is the identity matrix, and
2
λ,µ are the Lam´e constants, corresponding to the dilatation and shear modulus respectively,
given by
Eν E
λ = , µ = ,
(1+ν)(1−2ν) 2(1+ν)
where the Young’s modulus E captures the elastic stiffness in the direction of a load and the
Poisson’s ratio ν describes the stretch (compression) in the direction perpendicular to the
load. By introducing λ and µ, the effective stress is written in a symmetric form. To include
the effect of the fluid pressure inside the porous body, we introduce the addition stress due
to the pore pressure p :
p
p = αp, (1.4)
p
where the Biot-Willis constant α [23] satisfies α = 1−K/K with K being the bulk modulus
s
of the porous matrix and K is the bulk modulus of the solid material. In most situations
s
α ≈ 1, corresponding to an incompressible solid matrix. In a certain models of secondary
3